back to *Trigonometry*
# Analytical Geometry examples

### Example 1 of Ellipse Equation

### Example 2 of Ellipse Equation

### Example 3 of Circle Equation

- The standard form of an ellipse is:
- [(x - h)
^{2})/a^{2}+ [(y - k)^{2})/b^{2}= 1 - the center of the ellipse is (h, k)
- and its graph is:

- Question 1.
- Find the center of the ellipse given by the equation 9x
^{2}- 36x + 4y^{2}+ 24y + 36 = 0 and draw the ellipse.

- Solution:
- change the given ellipse equation into the stand ellipse equation
- 9x
^{2}- 36x + 4y^{2}+ 24y + 36 = 0 - 9(x
^{2}- 4x) + 4(y^{2}+ 6y) + 36 = 0 - 9[x
^{2}- 4x + (4/2)^{2}- (4/2)^{2}] + 4[y^{2}+ 6y + (6/2)^{2}- (6/2)^{2}] + 36 = 0 - 9[x
^{2}- 4x + 4 - 4] + 4[y^{2}+ 6y + 9 - 9] + 36 = 0 - 9[x
^{2}- 4x + 4] - 9 × 4 + 4[y^{2}+ 6y + 9] - 4 × 9 + 36 = 0 - 9[x
^{2}- 4x + 2^{2}] - 36 + 4[y^{2}+ 6y + 3^{2}]~~- 36~~+~~36~~= 0 - 9[x
^{2}- 4x + 2^{2}] + 4[y^{2}+ 6y + 3^{2}] = 36 - both side of the equation divide by 36 (each item divide by 36)
- 9[x
^{2}- 4x + 2^{2}]/36 + 4[y^{2}+ 6y + 3^{2}]/36 = 36/36 - [x
^{2}- 4x + 2^{2}]/4 + [y^{2}+ 6y + 3^{2}]/9 = 1 - [(x - 2)
^{2}]/2^{2}+ [(y + 3)^{2}]/3^{2}= 1 - the standard ellipse equation is: [(x - 2)
^{2}]/2^{2}+ [(y + 3)^{2}]/3^{2}= 1 - therefore, the center of the given ellipse is: (2 , - 3)

- The graphic of the ellipse is

- Question 2
- Find the center of the circle x
^{2}- 4x + y^{2}+ 2y - 4 = 0, draw the graph and find its area.

- Solution
- x
^{2}- 4x + y^{2}+ 2y - 4 = 0 - x
^{2}- 4x + (4/2)^{2}- (4/2)^{2}+ y^{2}+ 2y + (2/2)^{2}- (2/2)^{2}- 4 = 0 - x
^{2}- 4x + (2)^{2}- (2)^{2}+ y^{2}+ 2y + (1)^{2}- (1)^{2}- 4 = 0 - x
^{2}- 4x + 4 - 4 + y^{2}+ 2y + 1 - 1 - 4 = 0 - x
^{2}- 4x + 4 + y^{2}+ 2y + 1 - 4 - 1 - 4 = 0 - (x
^{2}- 4x + 4) + (y^{2}+ 2y + 1) - 4 - 1 - 4 = 0 - (x
^{2}- 4x + 4) + (y^{2}+ 2y + 1) = 4 + 1 + 4 - (x
^{2}- 4x + 4) + (y^{2}+ 2y + 1) = 9 - (x - 2)
^{2}+ (y + 1)^{2}= 3^{2} - therefore, the center of the circle is: (2, -1) and the radius of the circle is: 3
- The center of the circle is (2, -1), and the radius of the circle is r = 3
- The area of the circle is pi*r
^{2}= 3.14 * 9 = 28.26 square units.

- remark:
- circle equation: (x - x
_{o})^{2}+ (y - y_{o})^{2}= r^{2} - center of the circle is: (x
_{o}, y_{o}) and radius is r

© www.mathtestpreparation.com